The entanglement entropy is an interesting observable that quantifies the
long-range entanglement between classically separated quantum degrees of freedom.
It is a probe that can characterize the state of matter.
Calculations of the entanglement entropy rely on the determination of the
reduced density matrix for the degrees of freedom enclosed by an entangling surface,
by integrating out the degrees of freedom outside the surface.
This density matrix deviates from that of a pure state.
In this sense, there is an analogy with the thermal entropy, which
has been exploited in order to study the thermal nature of
field theories in curved spacetimes with horizons.
Explicit calculations of the entanglement entropy
are possible mainly in low dimensions
[1]}, [2]}, [3]}
and for non-interacting quantum field theories in higher dimensions [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}.
However, the holographic approach and the Ryu-Takayanagi proposal
[11]}, [12]} provide a simpler framework for theories that have a gravitational
dual in the context of the AdS/CFT correspondence [13]}.
The demanding calculation
of the density matrix is replaced by the simpler calculation of extremizing the area
of a surface that starts from the entangling surface on the boundary of AdS space
and extends into the bulk.
